with \(k^2=k_x^2+k_y^2\) . The bands are separated by two spectral gaps: \((0,f)\)  and \((-f,0)\) . It was shown in [1]}, [2]} that each band is associated to a well defined topological index, the Chern number, with respective values \(C_\pm = \pm 2\)  and \(C_0=0\) , attesting to non-trivial topology in the bulk (namely when \(\mathcal {H}\)  acts in \(L^2(\mathbb {R}^2,\mathbb {C}^3)\) ).
